DHAKA—Bangladesh's largest public protests in over two decades swelled, as tens of thousands of demonstrators took to the street Friday to demand the death penalty for people convicted of war crimes in the country's 1971 war of independence.

Meanwhile, clashes have intensified between security forces and Islamist protesters who claim Prime Minister Sheikh Hasina's government is using a war crimes tribunal it set up in 2010 to unfairly hound religious conservatives. On Thursday, two bystanders died when police fired rubber bullets on demonstrators, who were lobbing homemade bombs and vandalizing vehicles, police said.

Both sets of protests illustrate how the country remains polarized over the events 40 years ago, despite pledges from Ms. Hasina's secular-leaning administration the war crimes tribunal would be a model to the world and heal wounds of the civil war era.

A steady stream of people including middle-class families with children have flowed into Shahbag, a leafy boulevard in the heart of Dhaka, the capital, calling for those on trial for war crimes to be hanged. These protests, which so far have not come into contact with the Islamist-led demonstrations, have remained peaceful.